On Some Infinite Horizon Cheap Control Problems with Unbounded Data

Abstract: The definition ”cheap control problem” is used in the literature to denote a wide class of optimization problems, all having in common that there exists a set such that the system can move, in a neighbourhood of it, at a speed which is almost infinite with a negligible cost. In this paper we consider an infinite horizon cheap control problem, for a control-affine nonlinear system, with unbounded controls and a nonnegative vanishing lagrangian. As usual we embed the control set into an enlarged set of generalized or impulsive controls. We introduce sufficient conditions under which we prove the well-posedness of the generalized setting, i.e. that no Lavrentiev gap occurs, and show that the infima of various regularizations of the original problem converge to the infimum of it. These conditions are strictly related to some continuity properties of the infinite horizon value function V , which play a key role also for the characterization of V as maximal viscosity subsolution of a quasi-variational inequality.